# Homoclinism of Lie rings

## Definition

### Short definition

A homoclinism of Lie rings is a homologism of Lie rings with respect to the subvariety of the variety of Lie rings given by abelian Lie rings.

### Full definition

For any Lie ring $L$, let $Z(L)$ denote the center of $L$, $\operatorname{Inn}(L)$ denote the Lie ring of inner derivations of $L$ (explicitly, it is isomorphic to $L/Z(L)$, and $L'$ denote the derived subring of $L$.

Let $\gamma_L$ denote the mapping $\operatorname{Inn}(L) \times \operatorname{Inn}(L) \to L'$ that arises from the Lie bracket mapping $L \times L \to L'$, and then observing that this map is constant on the cosets of $Z(L) \times Z(L)$. Note that the mapping is $\mathbb{Z}$-bilinear.

A homoclinism of Lie rings $L$ and $M$ is a pair $(\zeta,\varphi)$ where $\zeta$ is a homomorphism of $\operatorname{Inn}(L)$ with <math\operatorname{Inn}(M)[/itex] and $\varphi$ is a homomorphism of $L'$ with $M'$, such that $\varphi \circ \gamma_L = \gamma_M \circ (\zeta \times \zeta)$.

## Related notions

Term Meaning
category of Lie rings with homoclinisms category whose objects are Lie rings and whose morphisms are homoclinisms
isoclinism of Lie rings an invertible homoclinism, or equivalently, a homoclinism where both the component homomorphisms are isomorphisms
homoclinism of groups the analogous concept for groups
homologism of Lie rings a more general concept of which homoclinisms are a special case. Homoclinisms are homologisms with respect to the subvarety of abelian groups in the variety of Lie rings.